getting 3 pocket Aces in a row

[BruceZ]

Another "exact approximation". I see. Well, I asked for something very specific, but I see that I cannot get through. The formula for "at least one streak of m occurences in n trials, when n is sufficiently large".

First of all, there are no "exact approximations". There are formulas which produce exact answers, there are formulas which can only produce approximate answers to within fixed upper and lower bounds, there are formulas which can produce answers to within any desired accuracy but never exact (infinite series), and there are formulas which can provide approximations to any degree of accuracy up to and including exactness. The formulas based on inclusion-exclusion I gave for opponents holding various hands such as AA when you hold KK are of this latter type. That is where this "exact approximation" stuff first came up. Those formulas could always be used to obtain an exact answer for any situation. We obtained an exact answer for AA vs. KK with 9 opponents, and we obtained exact answers for other hands with 4 players. For some hands with 9 players, we used the formula to produce approximate answers only, but these same formulas could have been used to produce the exact answers as well, it was simply not desirable to do so. What you seem to not understand is that the formulas derived by the inclusion-exclusion principle are exact when carried out to all the terms, and they represent closed form formulas. They are only approximate if you choose to make them approximate. The present case is another example, albeit one for which it would certainly be tedious to write out all the terms, and even more tedious to generalize it. I don't know why anyone would want to, but that doesn't mean it can't be done in theory.

In the end, everything is an approximation anyway because whatever you use to compute it on has finite precision, and the results are generally only used and only useful to very limited precision. So I think this obsession with exact closed form formulas is largely unjustified.

Your answer refers to boundaries. But this is not what I'm after.

Well excuse me! First of all, this thread began with a very specific question, not a general one. Failing a general solution, when someone does a lot of work to produce several very accurate approximations, the normal response is normally "thank you" , and not "your method is extremely inadequate" because it is only accurate to the zillionth decimal place. But that's fine, I don't spend my time posting solutions just for your approval.

Second of all, what you want in all likelihood does not exist anywhere. These renewal process problems are usually done with the Laplace transform methods like robk has presented, and these are also only carried out to approximation. Few texts on this subject would provide a solution to this problem based on simple combinatorics which I have provided. On this forum I have found a special case of a renewal process with p=1/2 that yielded a closed form solution in terms of the Fibonacci sequence. I have not found this solution described in the texts which deal with such processes, only the Laplace approximations. I am considering looking into whether my result is generally known, and I am also working on generalizing it.

Third of all, I gave you an exact closed form formula in terms of conditional probabilities. If you're too lazy to compute all the terms that isn't my problem.

I made very clear in my post that we could obtain an answer using the BinomDist formula if we want the probability of getting exactly one streak of exactly 3 Aces in a row in the next n hands. You are supposed to prove that this is "nonsense" or "terrible analysis" using mathematics. And not your kind of "congenial" diversions.

But you have not shown how this probability can be computed exactly with the binomial distribution, nor can you, because this probability is not binomially distributed. I wrote a whole paragraph, of which you only made fun of the length (did you bother to read it?) where I detailed the reasons why this is the case. What BinomDist gives you is the probability of exactly 1 success of probability p in n independent bernoulli trials, or at most 1 success, or at least 1 success if you subract the result from 1. We don't have independent bernoulli trials, so the whole thing is irrelevant to your analysis except as an approximation, and I know you don't want an approximation, so why you continue to push the binomial distribution is completely beyond me. I don't know what you intend to do with it. If you are going to make a claim that you can use this distribution to compute this probability, you had better damn well show how it can be done. If you say you can square the circle, the onus is not on me to prove that you can't.

Both AC_Player and myself have corrected you that if you do use the binomial distribution to approximate this, you can get the probability of at least 3 in a row also. Furthermore, I also indicated why the sum of these exact probabilities you indicated do not produce the correct probability. In the words of the late Peter Griffin "I gave you an explanation. I don't owe you an understanding".

The formula for the problem I have posited is not an "elementary" one. But from now on I will respectfully and pre-emptively accept that you understand everything and that you have the answers for everything.

But the specific problem we were solving (at least 1 run of length 3 or more in 250 hands) lends itself to extremely simplistic methods which are extremely accurate. I completely understand the problem, the limitations of the simplistic methods when applied to the problem, and the errors in the analysis which you continue to advocate. Do you?

I do not recall being anything but congenial, at least in this forum. But if your idea of congeniality is the tone and the attitude you have adopted throughout this thread, then I'm sorry but you have no idea what the word means.

No one spends more time and energy than I do helping people on this forum. I get pms all the time from people who really appreciate it. Some people don't feel they know enough to ask questions publicly, so they learn by reading or asking questions privately. I'm more than generous with my time, and have almost infinite patience with people of all levels of competence. You've asked some questions, and I've been particularly generous trying to help you. However, my patience quickly runs out when my efforts are unappreciated, misunderstood, and even ridiculed. I will always challenge incorrect ideas masquerading as truth. Normally they are quickly reasoned out to a conclusion, and not irrationally defended to the death. You need to be able to separate criticism of ideas from personal criticism. My characterization of what you wrote was accurate, and it becomes more accurate as you continue to defend something which is blatantly wrong. Your characterization of what I wrote was not accurate.

[Cyrus]

"First of all, there are no exact approximations."

You're telling me. I used your term, only to show how preposterous the concept is. You see, I had asked in a previous thread for an exact answer and you insisted that you gave one but that it differed from the exact one by "a small amount". And this I'm supposed to "appreciate" and stop "pestering you".

Indeed, this is an infrequent but bothersome problem with your posts, or at least the ones I have perused the short time I have : You are very cavalier with terminology!

And that comes across despite your sincere willingness and the effort you put in trying to help others, in cases like the wild Goedel misconcenptions (comingling the statement "P is neither true nor false" with the statement "P cannot be proven to be true nor false"!) or your recent misuse of the term Expected Value ("Suppose you play three 4 hour sessions. In the first session you win $200. In the second session you win $400. In the third session you lose $300. Your average win or EV is = $100/session or $25/hour.". No, Bruce, this is not "EV". This is just Average Win. Your indeed very thoughtful and helpful post could create such a confusion in the entry level player's mind about what EV is, that the worth of your post diminishes sharply.)

Do not, please, mistake my unwavering insistence for clarity and exactitude for anything but that. Calling this attitude names ("argumentative troll") is characteristic of neither a mathematician nor an aspirant teacher.

"No one spends more time and energy than I do helping people on this forum."

I have not disputed this. As a matter of fact I have given you a very high contributor's rating for precisely those efforts of yours. I for one appreciate them.

So, a parting word, which is not original and has already been suggested to you, in paraphrasing : Being precise does not have to lead to significantly long or less comprehensible posts.

Take care.

--Cyrus

[BruceZ]

or your recent misuse of the term Expected Value ("Suppose you play three 4 hour sessions. In the first session you win $200. In the second session you win $400. In the third session you lose $300. Your average win or EV is = $100/session or $25/hour.". No, Bruce, this is not "EV". This is just Average Win. Your indeed very thoughtful and helpful post could create such a confusion in the entry level player's mind about what EV is, that the worth of your post diminishes sharply.)

No Cyrus, you are wrong, but I'm glad you brought it up, and I will clarify this on the other forum as well. I don't know if I created any confusion, but your attempt to be pedantic here may very well have done so, but that's OK because I will clear it up now to everyone's benefit. I'm not going to debate semantics, though my semantics are technically correct, because what is important are not semantics but the understanding one has of concepts, and how to correctly apply them in practice. Inability to do this effectively is a common shortcoming of many bookish types and mathematicians who only understand theory, as opposed to those with a more practical bent, such as those with a background in engineering, physics, or statistics. At the same time, some of these practical folks may have lapses in fundamental theory. Different fields even use the same terms to mean slightly different things, the notion of independence being one example. My own background is in all these fields, so I will attempt here to bridge the gap.

EV stands for expected value. An elementary statistics book will tell you that the terms "mean", "average", "expectation" and "expected value" are all synonymous. One may compute a mean, average, expectation, or expected value over a finite number of samples, as I have done in my example. Now it is true that the term expected value is normally used by mathematicians to describe the mean of a random variable or of a probability distribution. As such, it can only be obtained from an infinite number of samples of the random variable, or in the limit as the number of samples approaches infinity by the law of large numbers.

When a gambler says "my EV is $50/hr", this can mean one of two very different and equally valid things. He can mean that he feels that the game he is in worth $50/hr to him, and if he plays until infinity under identical conditions, this would be his average win per hour. How can he know this EV? After all, nobody can really play to infinity. If the game were a relatively simple one like blackjack, he would simply need to perform a mathematical calculation or simulation taking into account the rules, number of decks, deck penetration, bet size, and strategy. If the game were more complex like poker, perhaps he made some estimates based on how he perceives his skill relative to that of his opponents, and how much that should be worth in dollars. Perhaps, on the other hand, he did not perform such a calculation, but instead simply averaged his actual winnings over some number of hours of play, as I did in my example. If he played for a very long time, the law of large numbers tells us that his result should be very close to the EV he would obtain if he had played until infinity. If he didn't play for such a long time, his result could differ significantly from this value, but it would still be an average of his results nonetheless, and he would use this value to estimate the EV he would obtain if he were to play forever. My simple example used only 3 sessions for simplicity, but the method was meant to be illustrative of how one would perform this computation over many more sessions, in which case one would arrive at a number very close to the EV if one played forever.

Hence there are two different EVs at play here, one which is a theoretical result that he would obtain if he could play forever, and this is an inherent characteristic of the player in the game. The other is a practical result that he computes after a finite number of hours. The latter of these is the more practical notion of EV for complex games such as poker, since the other one can never be known precisely. The term "sample average" is often used to describe the average of a finite number of samples, and the sample average is an estimator of the long term EV.

Just as an average can be computed from a finite number of samples, a standard deviation can be computed from a finite number of samples too. This is what I did in my example. This is a "sample standard deviation", and it is an estimate of the theoretical standard deviation one would obtain if one played forever. Perhaps you have a calculator (or Excel) which gives you the mean and standard deviation of a series of numbers. You don't seem to have a problem calling this a standard deviation, it even says standard deviation next to the key. You should not have a problem calling the mean an EV either, though I admit this is somewhat atypical and potentially confusing. If you study statistics you learn that variance, which is the square of the standard deviation, is the “second moment” of a generating function when the mean is taken to be zero. The first moment is expected value. If you don't understand this that's OK, my intent is simply to link the notions of EV and standard deviation.

Actually the question of whether EV and standard deviation must be regarded as fixed numbers inherent to a situation, or if they can be regarded as probability distributions whose true value can never be known but only estimated from finite information is a debate which gives rise to different branches of statistics and estimation theory. It is tied to the question of the very definition of probabillity, and whether probability is an inherent quantity, or one which can only be obtained through experiment. Your argument that an average win is not an EV is tantamount to arguing that probabilities can never be known, and are simply arbitrarily assigned quantities, rather than experimentally measured ones. You would not be alone in making this argument, but that does not make the practical argument incorrect or invalid. This is also related to fundamental questions regarding the apparent probabilistic nature of the universe as described by quantum mechanics. I'm digressing to indicate that this issue actually runs much deeper than our discussion.

"First of all, there are no exact approximations."

You're telling me. I used your term, only to show how preposterous the concept is. You see, I had asked in a previous thread for an exact answer and you insisted that you gave one but that it differed from the exact one by "a small amount". And this I'm supposed to "appreciate" and stop "pestering you".

I used this term as a title to a post in a humorous sort of way in response to your question, which was something like, "how can something be both exact and approximate". I feel that you still do not appreciate this concept, and that this is a serious shortcoming which underlies your inability to extract value from this thread as well. The fact is that I GAVE AN EXACT ANSWER TO THE PROBLEM IN QUESTION. Not just an approximation, not an exact approximation, but a genuine exact answer. I did it in an earlier thread, and I did it very concisely at the bottom of the thread you referenced in the post titled "Much simpler way!" The whole purpose of that discussion was to show that one may begin with a very simple first-order approximation (do you know what that is?) which is often very accurate in itself, and then continue to refine this by adding higher order terms until one eventually arrives at the exact answer, and this can always be done for problems with a finite number of outcomes. THIS IS AN EXTREMELY POWERFUL AND IMPORTANT CONCEPT. You will not find it described in this kind of detail in very many places. In the thread that you referenced I provided more explanations to you than most anyone would require, and still you failed to grasp the importance of this. Sorry if this sounds harsh, but this makes it obvious that you possess no qualification to evaluate my explanations of anything of a mathematical nature, NONE WHATSOEVER!

in cases like the wild Goedel misconceptions (comingling the statement "P is neither true nor false" with the statement "P cannot be proven to be true nor false"!)

I never claimed these statements are equivalent. What I said was "It may be a surprise to a lot of people to learn that not all things are either true or false". That is a true statement! Then I said "There are some things that are neither true nor false, but in a third state of 'undecideable', and you can prove that". That is also true in the field of quantum mechanics, and I had this in mind when I wrote that post. This has nothing to do with Gödel. The purpose of these statements was to spark controversy and debate, and to set up the refined explanation which followed.

I have freely admitted that I was very sloppy in the terminology I used later in that post when discussing Gödel’s theorems, and I subsequently posted a multi-page post which corrected this and explained the matter clearly.

Calling this attitude names ("argumentative troll") is characteristic of neither a mathematician nor an aspirant teacher.

Now let's not commingle comments made in jest in a forum intended to let one's hair down and blow off steam, with statements made on a math forum. My comments here are deadly serious. If I see something that is garbage, I will call it garbage. If you can't handle that, then don't post on a math forum.

Being precise does not have to lead to significantly long or less comprehensible posts.

True, though there is compromise between quantity of information conveyed, time spent, and the probability that one may occasionally be imprecise. This is one reason why twoplustwo should be commended for maintaining both high quality and high quantity of information it publishes. That isn't an easy thing to do. I think I negotiate this compromise pretty well, and I do try to always be precise, and I was precise in two of the three examples you cited here. It is my hope and belief that other's agree. If they didn't, then I would not post nearly as much.

Responses from others are welcome, and from you as well even though you said it was your final post.